Solve the exponential equation for $x$. 49 − x − 7 ⋅ 7 4 x + 8 = 49 5 x − 3 49\^{-x-7}\cdot 7\^{ 4x+8}=49\^{ 5x-3} $x=$
Solution: The strategy Let's write $7$ in base $49$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $49$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 49 − x − 7 ⋅ 7 4 x + 8 = 49 − x − 7 ⋅ ( 49 1 2 ) 4 x + 8 = 49 − x − 7 ⋅ 49 2 x + 4 = 49 − x − 7 + ( 2 x + 4 ) = 49 x − 3 ( 7 = 49 1 2 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 49\^{-x-7}\cdot 7\^{ 4x+8}&=49\^{-x-7}\cdot (49\^{ \frac12})\^{ 4x+8} &&&&(7=49\^{ \frac12})\\\\ &=49\^{C{-x-7}}\cdot 49\^{ {2x+4}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=49\^{ C{-x-7} \ + \ ({2x+4}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=49\^{ x-3} \end{aligned} Solving the linear equation We obtain the following equation. 49 x − 3 = 49 5 x − 3 49\^{ x-3}=49\^{ 5x-3} Now we can equate the exponents and solve for $x$. $\begin{aligned} x-3 &=5x-3\\\\ x &= 0\end{aligned}$ The answer The answer is $x=0$. You can check this answer by substituting $\it{x=0}$ in the original equation and evaluating both sides.